Concepts Simplified: What are Quadratic Equations?

What is a Quadratic Equation?

It’s just a fancy name for an equation like this: $x² + 5x + 6 = 0$
Or more generally:
Standard Form: $ax^2 + bx + c = 0$
It ALWAYS has an $x^2$ term — that’s what makes it “quadratic.”

But What Are We Trying to Do?

We want to find the value of $x$ that makes the left side equal to 0.
Think of it like solving a mystery number that fits the equation.

Let’s Start with a Simple One

$x^2 + 5x + 6 = 0$

We want to find two numbers that:
1. Multiply to give 6
2. Add to give 5

2 and 3

$2 \times 3 = 6$, and $2 + 3 = 5$

So we factor:
$(x + 2)(x + 3) = 0$

Then use a basic math rule:
If $(A)(B) = 0$, then either $A = 0$ or $B = 0$

So:
- $x + 2 = 0 \Rightarrow x = -2$
- $x + 3 = 0 \Rightarrow x = -3$
These are your solutions!

When It’s Not So Easy...

Try this: $2x^2 + 7x + 3 = 0$
This doesn’t factor easily, so we use the formula.

The Magic Formula

For any quadratic equation $ax^2 + bx + c = 0$, the solution is:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Let’s solve $2x^2 + 7x + 3 = 0$
- $a = 2$
- $b = 7$
- $c = 3$

Plug into the formula:
$$ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2} $$
$$ x = \frac{-7 \pm \sqrt{49 - 24}}{4} = \frac{-7 \pm \sqrt{25}}{4} $$
Since $\sqrt{25} = 5$:
$$ x = \frac{-7 + 5}{4} = \frac{-2}{4} = -0.5 $$
$$ x = \frac{-7 - 5}{4} = \frac{-12}{4} = -3 $$

Visual Analogy

Think of $x$ as a hidden block. You're building a puzzle with $x^2$, $x$, and number pieces — and you're arranging them so the total equals zero.

Practice for Confidence

Try solving these:
1. $x^2 + 6x + 9 = 0$
2. $x^2 - x - 12 = 0$

Final Takeaway

- Don’t fear $x^2$ — it’s just $x$ times $x$
- Try factoring first
- Use the magic formula when stuck
- Solving just means: “What value of $x$ makes this equation zero?”